1.2. Lecture 2
1.2.1. More results about the exponential map
For the next result about the exponential map it will be useful to know the following fact from linear algebra.
Lemma 1.2.1.
Let
Proof.
(nonexaminable) This follows from Jordan normal form. Here’s a direct proof. Firstly write
and by the rank-nullity theorem
is a decomposition of
Lemma 1.2.2.
The exponential function
Proof.
First prove it for all
Now consider
because
Remark 1.2.3.
The lemma is not true over
Lemma 1.2.4.
We have
Proof.
By Lemma 1.2.1 we can conjugate so that
Thus
∎
The next proposition will be useful when we discuss Lie algebras of linear Lie groups.
Proposition 1.2.5 (Lie product formula).
For
Proof.
We have
Now take the limit as
1.2.2. One-parameter subgroups
Lemma 1.2.6.
The map from
is a differentiable group homomorphism.
We have
In particular,
Proof.
The given map is a group homomorphism by Lemma 1.1.2(iv).
By definition,
As this power series (and its termwise derivative) are uniformly convergent on any compact subset, we can compute its derivative by differentiating termwise, which gives
Definition 1.2.7.
A one-parameter subgroup of
for all
The infinitesimal generator of a one-parameter subgroup
The convention used here is a slight abuse of notation,
Remark 1.2.8.
For a one-parameter subgroup
Indeed, if
The RHS is differentiable with respect to
It is well-defined for
The following is a very important property of one-parameter subgroups: that they all come from the exponential map.
Proposition 1.2.9.
Let
Then
for all
Proof.
From the definition of one-parameter subgroups, we have
Now consider the differential equation
By Lemma 1.2.6 we have
and thus
Example 1.2.10.
The map
1.2.3. Exercises
Problem 3. Let
In this problem we will see that the restriction of the exponential to
-
(a)
Let
. Show that . -
(b)
Show that
for . -
(c)
Show that, for
, the logarithm is in fact a finite sum (and hence converges). -
(d)
Show that
and are inverses of each other. Hint: this boils down to an identity of formal power series, which you can actually deduce from the corresponding fact over .
Problem 4. Using the previous question, fill in the gaps of the proof from the notes that
is surjective.
-
(a)
Show all matrices of the form
have a preimage. -
(b)
Show all matrices of the form
have a preimage. -
(c)
Show all matrices of the form
have a preimage. -
(d)
Is the exponential map
surjective? What about ?
Problem 5. Let
Show that